As climate change looms larger, many look to sustainable investing that incorporates environmental, social, and governance (ESG) concerns as part of the way forward. To assess scope for ESG-conscious investing to achieve climate change goals, we explore the link between emissions growth and ESG scores using firm-level data for the largest emitters around the world. Discouragingly, our analysis uncovers at best a weak relationship: firms with better ESG scores do display somewhat slower emissions growth but this link is substantially attenuated and no longer statistically significant if we limit attention to within-country or within-firm variation. Our findings suggest limited scope for sustainable investing strategies conditioned solely on ESG indicators to meaningfully help mitigate climate change and, more broadly, underscore the need to continue to build consensus towards effective economy-wide policies to address climate change.